Notes of Enrico Le Donne’s lecture

Sub-Riemannian geometry working seminar, organizational matters

– We meet every wednesday, 5pm, at IHP, amphi Darboux (except for a few exceptional days where we move to room 05, same floor). If people prefer a different schedule, let me know.

– People welcome to join for dinner after seminar.

– Many people have offered to give a talk. After today’s talk, we shall sort out proposals.

1. Metric spaces not parametrized by their tangents

1.1. Sub-Finsler manifolds

Definition 1 An (equiregular) sub-Finsler manifold is the data of a Finsler manifold (a smoothly varying norm is given on each tangent space) and a smooth sub-bundle {\Delta\subset TM} satisfying the following assumptions. Let {\Xi^k} denote the family of vector fields obtained iteratively from {\Xi^1 =} smooth sections of {\Delta} by the Lie bracket operation,

\displaystyle  \begin{array}{rcl}  \Xi^{k+1}=[\Xi^1,\Xi^k]. \end{array}

Let {\Delta^k =\Xi^k (p)} denote the set of values at point {p} of elements of {\Xi^k}. Then we assume that

  1. There exists {s} such that {\Delta^s =TM}.
  2. All {\Delta^k} are sub-bundles.

A sub-Finsler manifold comes up with a distance: given two points, minimize length of curves between them which are almost everywhere tangent to {\Delta}. Note this distance only depends on the restriction of the norm to {\Delta}. So in fact one does not need the norm is other directions. It is a leitmotiv of sub-Finsler geometry: although, occasionnally, we may use geometric data defined outside {\Delta}, what really matters for us is {\Delta} and the associated metric.

1.2. Carnot groups

Carnot groups form an important class of examples.

Definition 2 Let {\mathfrak{g}} be a Lie algebra. Say that a direct sum decomposition {\mathfrak{g}=V^1 \oplus\cdots\oplus V^s} is a stratification of {\mathfrak{g}} if

  1. {[V^i ,V^j]\subset V^{i+j}}.
  2. {V^{k+1}=[V^1 ,V^k]}.

Pick a norm on {\mathfrak{g}}, left-translate it, this gives a sub-Finsler manifold.

Example 1 {\mathfrak{g}=V^1 \oplus V^2} with {V^1={\mathbb R}^2}, {V^2={\mathbb R}} and {[\cdot,\cdot]:V^1 \times V^1 \rightarrow V^2} is the determinant. This is called the first Heisenberg Lie algebra.

Picture…

1.3. Tangent cones

Theorem 3 (Mitchell 1985, completed by other people) The tangent cone of a sub-Finsler manifold et a point {p} is a Carnot group (which may depend on {p}).

Here, tangent cone of {M} at {p} means the limit of dilates of {M}. Limit is taken in Gromov-Hausdorff sense.

Definition 4 Let {X}, {Y} be bounded metric spaces. Define

\displaystyle dist_{GH}(X,Y)=\inf dist_{H}(X'\subset Z,Y'\subset Z),

where inf is taken over all metric spaces {Z}, and all subsets {X'\subset Z} isometric to {X} and {Y'\subset Z} isometric to {Y}, and maps preserve base points.

For unbounded pointed metrc spaces {(X_n,x_n)} and {(Y,y)}, say that {(X_n,x_n)} tends to {(Y,y)} if for all {R>0}, the {R}-balls converge, i.e.

\displaystyle  \begin{array}{rcl}  dist_{GH}(B^{X_n}(x_n,R),B^{Y}(y,R))\rightarrow 0. \end{array}

Here we are in fact dealing with (pointed) metric spaces up to isometry.

Metric spaces need not admit tangent cones. The doubling condition is a favorable circumstance: it guarantees compactness of dilates, and so existence of converging subsequences among dilates. Nevertheless, limits need not exist (i.e. sub-limits need not be unique), even in the class of doubling metric spaces.

Theorem 5 (Le Donne) Let {X} be a geodesic metric space. Assume {X} has a doubling measure {\mu}. Assume that {\mu}-almost everywhere, {X} has only one tangent cone (i.e. dilates converge to some metric space). Then the tangent cone is a Carnot group {\mu}-almost everywhere.

The proof uses the following

Lemma 6 In a doubling metric space {X}, almost every point {x} has the following property. Assume {(Y,y)} is a tangent cone to {X} at {x}. Then for all {y'\in Y}, {(Y,y')} is again a tangent cone of {X} at {x}.

and relies essentially on the following

Theorem 7 (Gleason, Montgomery, Zippin 1950, Berestovskii 1988) Assume {Y} is a complete, proper, connected and locally connected metric space. Assume isometry group is transitive. Then {Isom(Y)} is a Lie group with finitely many connected components.

Moreover (Berestovskii), if {Y} is geodesic, then {Y} is a sub-Finsler manifold.

1.4. Impossible parametrization by tangents

For sub-Riemannian manifolds, the tangent space is not a local model in general.

Theorem 8 (Le Donne, Ottazzi, Warhurst 2011) There exists a nilpotent Lie group equipped with a left-invariant sub-Riemannian metric which is not locally quasiconformal to its tangent cones.

Note that bi-Lipschitz {\Rightarrow} quasiconformal.

Definition 9 Say a Carnot group is ultra rigid if every even locally defined quasiconformal self-mapping is the restriction of the composition of a translation and a dilation.

Theorem 8 easily follows from the following

Lemma 10 There exists a nilpotent non Carnot Lie group {H} equipped with a left-invariant sub-Riemannian distance for which the tangent cone {G} is ultra rigid.

Indeed, ultra rigidity provides an injective group homomorphism from {H} to the group {G\times\mathbb{R}} of translations/dilations of {G}. The semi-direct product is not nilpotent, so homomorphism is not onto (at the level of Lie algebras), {\mathfrak{h}} maps injectively to {\mathfrak{g}}. Since they have the same dimension, {\mathfrak{h}=\mathfrak{g}}, contradiction.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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